This mathematics is involved with kissing numbers. The big sourcebook on this is Conway and SLoane, "Sphere Packing, Lattices and Codes." These spheres in the 4-dim case are connected to Planck units of volume. The packing system is the 24-cell or equivalently the F4 group.

These systems are error correction codes. In the case of sporadic groups the quantum error correction is meromorphic which preserves quantum information. In the simple case there is no pole this recovers unitarity. In the study of this it is important to keep the connection to Jacobi theta functions. This also connects up with the Ramanujan Mock theta function and the partition function for the integers. This partition function is related to the density of states of a bosonic string as well as the thermal partition function of a black hole.

There are deep relationships involved with this. My essay here is an attempt to lay down some physical arguments for this.

Cheers LC

Dear Lawrence,

Congratulations on a well written essay dense with impressive references to many relevant issues raised by the fqxi contest question. The technical aspects of your discussion went over my head (and probably for many others here), particularly in the section about logic and in the applicability of the Incompleteness Theorem to the issues at hand.

That said I could confidently say that I agree with several of your points: 1- The need for a 'philosophy' to approach questions of Reality in physics. 2- The undecidability of It/Bit 3- That a density matrix allows the expression of quantum states as qubits (which was my conclusion). In my Theory (see below) GR is reduced to a density gradient. 4- That "in theoretical physics there may exist assumptions that act as excess baggage that prevent workers from addressing fundamental problems" which was a major argument in my paper, although your saying it sounded much less presumptuous than when I did. Not only in this contest, but in my last year's "Fix Physics!" essay - since most of my ideas are qualitative. 5- The relevance of causality sets which in my Beautiful Universe Theory also found here are simply the Hamiltonians of qubit-like (spherical degree of freedom at every point) transfer of angular momentum locally, causally and linearly in a Universal lattice.

On a personal note I visited Purdue around 1965 to visit my brother-in-law who did his PhD in physics there. One is wont to believe in a Flat Universe in that locale!

With best wishes for your success

Vladimir

    Hi Vladimir,

    Indiana is not just a case of a flat universe, but a flat Earth. It is in many ways socially backwards, and there were Ku Klux Klan rallies in the 1980s when I was there. That is rather embarrassing.

    The elimination of excess baggage is important. In my elementary demonstration with modal logic it means that certain physical axioms or postulates can be "turned off" in certain domains. It is similar to Godel's theorem, where certain propositions about a mathematical system are not provable and they can be toggled on or off to create different systems. Euclid's fifth axiom is of that nature and its on and off state define euclidean flat geometry and Riemannian geometry respectively. Your FQXi essay reads a bit like a narrative on this sort of thing.

    Cheers LC

    Darrell,

    Your essay does read as some narrative on a similar idea. You take the perspective of an alien. It will be extremely interesting if we should ever get radio contact from ETI to see how different they perceive the universe. The question is whether their mathematics and physics are in some ways mapped into ours, or if there is some isomorphism betewen their math and physics and ours.

    Cheers LC

    Hi Lawrence,

    I agree that ""It From Bit" is not decidable", or rather, that it is a question which belongs to what to me is an outdated paradigm.

    I have yet to read an essay which treats the question where all information comes from, how information becomes information. What I mean is this: If there would be only a single charged particle among uncharged particles in the universe, then it wouldn't be able to express its charge in interactions. As it in that case it cannot be charged itself, charge, or any property, for that matter, must be something which is shared by particles, something which only exists, is expressed and preserved within their interactions. If particles, particle properties (its) are both cause and effect of their interactions, of the exchange of bits, if particles only exist to each other if, to the extent and for as long as they interact, exchange information, then you cannot have one without the other nor can one be more fundamental than the other.

    If the information as embodied in particle properties and the associated rules of behavior a.k.a. laws of physics must be the product of a trial-and-error evolution, then information only can survive, become actual information when tested in practice, in interactions between its carriers, between actual, physical, material particles, whatever we may mean with 'material'.

    What strikes me in all the essays I've read (also of previous contests) is that everybody, without exception, thinks about the universe as an object which has particular properties as a whole and evolves in time, as something we may imagine to look at from the outside and can make statements about, as if it is a mechanism which, once winded up at the bang, after the bang only can unwind in a preordained fashion, as if there is a collection of platonic truths which exist outside the universe, as if there is an absolute, objectively observable reality at the origin of our observations we cannot perceive due to imperfect instruments and to the uncertainty principle.

    My point is that if a particle cannot exist, have properties if there's nothing outside of it to interact with, then the same must hold for the universe. The fallacy of Big Bang Cosmology (BBC) is that we can only speak about the properties and state of the universe if there's something outside of it, something it can interact with, and, like the charged particle its charge, something it owes its properties to: if it has been created by some outside intervention. For this reason BBC is an even worse 'theory' than creationism which at least honestly states that, yes, there is Someone outside of it Who created the universe. If a universe which creates itself out of nothing, without any outside intervention has to obey the conservation law which says that what comes out of nothing must add to nothing, then everything inside of it, including space and time somehow must cancel, add to nil, meaning that it has no physical reality as a whole, as 'seen' from the outside, but only exists as seen from within. If in that case it doesn't make sense to speak about the properties it has or the state it is in as a whole, then it also makes no sense to make such statements from within. In other words, we need a completely different approach, an entirely different paradigm if we ever are to comprehend the universe rationally, as opposed to causally, something I'm trying to do in blog, a study which, I'm afraid, is a bit of a mess.

    As I argued in a previous essay, this means that we can no longer conceive of the speed of light as the (finite) velocity light moves at, but that c just refers to a property of spacetime, which is something else entirely. (Though you agreed with my statement that we ought to "replace causality with reason ...to understand our world", I hope that you realize, accept that if causality goes out the window, then so does the interpretation of c as the (finite) velocity of light.) In regarding the universe as an object we can imagine to look at from without, a Big Bang Universe (BBU) lives in a time realm not of its own making: as it is the same cosmic time everywhere, here it takes a photon time to travel so here c does refer to the velocity light moves at. In contrast, a Self-Creating Universe (SCU) does not live in a time realm not of its own making: as it contains and produces all time within, here clocks are observed to run slower as they are more distant even if they are at rest relative to the observer. As in a SCU it is not the same time everywhere, here a space distance is a time distance so in this universe a photon bridges any spacetime distance in no time at all, in contrast to a BBU where the photon covers a space distance in (a finite) time. The difference is as subtle as it is crucial to comprehend our universe. Evidently, in a universe where the communication between particles over any spacetime distance is instantaneous, things like the double-slit experiment, the EPR paradox become obvious. The problem is that nobody seems to be able to escape the essentially religious narrative of BBC and start to try to understand the universe from within. Frankly, I'm appalled that everybody takes the word of the heroes of physics as a God's word instead of trying to see whether a different interpretation of observations might solve some of the most glaring contradictions of physics.

    Regards, Anton

      Undecidability is a universal property, such as it is known that not all propositions in Peano arithmetic are decidable. The Church-Turing thesis states that any computable function is run on a Turing machine that halts. Of course it is known that a universal Turing machine is not capable of determining whether all TMs are halting. Hence the λ calculus is incapable of verifying Hilbert's Entscheidungsproblem.

      My paper works with certain correspondences between modal logic, Lob's theorem and the Godel theorem. I did not break this out in great detail to avoid turning the paper into a complicated discussion of symbolic logic. To address your question again; I think what you are asking is how can we determine what is undecidable. This is an area of research for people who delve into this subject. There are mathematics on proof theory and hierarchies of undecidability. I will confess that I am not an expert in this area of work, for this deviates seriously from physics.

      My argument is then more physical than formal, where I consider a certain algebraic postulate about observables, that being associativity. The physical argument involves observables or quantum amplitudes in a region that contains a black hole.

      Cheers LC

      Anton,

      I am sorry I can't respond in the length you write. The day is getting a bit on already and I need to attend to other things. There are a couple of points I can make.

      If the universe had one charged particle something funny would happen. If we consider the spatial surface of the universe, say on the Hubble frame, as a 3-sphere the lines of electric force from this particle would wind around this sphere endlessly. Indeed these lines of force would densely fill the space. They would also interact with themselves or in effect the charged particle. The charged particle would be "driven" into a divergent condition to rapidly increasing energy. The system would in effect "explode." It is comparable to an undamped oscillator that is driven by a resonant frequency.

      The recent developments with inflationary cosmology now involve bubble nucleations or O-regions that result from a vacuum transition to a small value. This is an aspect of the multiverse. There is some prospect that our O-region, bubble or sometimes called pocket universe interacted with another one in its early phase. This may have left an imprint on the CMB. The de Sitter spacetime where this inflationary expansion occurs may in turn be a vacuum-field configuration on a D-brane, where in this higher dimensional space of 10 or 11 dimensions there is a foliation of such branes. This then leaves the bulk 10/11 dimensional spacetme, which might in turn exist in a 26 dimensional spacetime and so forth. This sort of gets into the proverbial idea, "its turtles all the way down."

      Cheers LC

      Dear Crowell,

      Now let us call the most natural automaton simply "heredity" (or indeed any cycle, Carnot or whatever). Then the idea of entropy or 2nd law of thermodynamics is that each stage of evolution within an automaton path does in fact depreciate possibility of return to the initial state; so there must be a halt or "fatalism"

      My question is: doesn't your undecidableness amount to what we already know as uncertainty? Such that above a cut-off (Heisenberg Cut?) there is higher probability of return to initial state (what we know as determinism) but below the cut (being what we know as the quantum scale) there is no probability of return to initial state i.e. the system is not well-behaved.

      Now I put it to you that once you assume that a well-behavedness (or conversely "uncertainty") is simply put an automaton then it qualifies as the ALGORITHM or computer proper. This computer/algorithm is what I call simply THE OBSERVER (perhaps Wheeler's anticipation).

      It is a response to your position that: "...a quantum field is propagating on spacetime,but where spacetime is the quantum field. This heuristically appears self-referential, and the physical ansatz is this nonlocality is an undecidable proposition of the above modal theory of causality."

      By quantum theory, this state is rather at once a Godel "incompleteness" (the uncertainty) and yet a Peano (or Planck) "constant" (i.e. the conservation law).

      My "observer" is in other words then the SUPERPOSITION proper.See What a Wavefunction is

      So Wheeler's proposition does work! Pls see my essay and prove me wrong on this particular approach.

        I hesitate to call the Heisenberg uncertainty principle some derivative of Godel's theorem. However, my ansatz that this incompleteness rests with the associativity of fields is really just another form of the quantum uncertainty. There are limited ways of knowing what propositions about a mathematical system are undecidable, so my assumption here is not a form derivation of any sort. So my idea here is more of a physical assumption than a formal proof or derivation.

        The role of an observer is in some ways similar to a self-reference in mathematics. Godel's trick is to let a mathematical system contain predicates that act on their own Godel numbers as the subject. An observer is ultimately made of quantum particles and the act of a measurement has the appearance of being a sort of physical form of self-reference. Of course my paper is not about the measurement problem or the role of observers, but ultimately the universe is as it is so that observers can exist. This is a sort of strong form of the anthropic principle.

        As for fatalism, it is of course the case that the universe will end up as a pure de Sitter vacuum in a sort of heat death. This will be reached in around 10^{110} years. It will probably quantum decay from there over a far longer period of time into a Minkowki vacuum. That might sound like fatalism, just as saying that we are all going to die sounds fatalist. However, in the mean time a lot of different things can happen. In the sense of Plato's final cause the outcome might be in a sense fated, but what happens before then is not.

        Cheers LC

        Lawrence,

        There are two kinds of scientists, the kind which can perfectly build further on the theories one learned at school, and the kind which tries to find alternative interpretation of observations. However invaluable an education is, its disadvantage is that, since you've learned many theories describing phenomena, it is very hard to dream up a different approach which perhaps might solve some of the many fundamental contradictions and enigmas of present physics. Though it is very understandable that man came up with the big bang idea, I'm afraid that it actually is even a worse 'theory' than creationism which at least honestly, boldly states that the universe has been created by some outside intervention. As far as I am aware of, the universe either has been created by some outside interference (which is the position of both creationism and big bang cosmology), a possibility I reject or it creates itself. If so, then fundamental particles must be as much the cause, the source of forces as their effect, the product of their interactions, meaning that a force cannot be either attractive or repulsive, always, which is the mistaken belief string theory is based upon. String theory, like big bang cosmology don't solve anything but are part of the problem.

        Cheers AB

        6 days later

        Dear Lawrence

        Your quote from comment to Platan essay:

        "What do you think of algebraic curves over [0, 1, ∞] and the Langlands program?"

        By coincidence i used long time ago similar trick with pseudoscalar mesons.See my essay. I signify as 1 mass of proton and then observed what position would take place other pseudoscalar mesons.That i got phenomenon 18 degrees.

        Do you see some explanation?

        Yuri